## Taiwanese Journal of Mathematics

### MULTILINEAR ESTIMATES ON FREQUENCY-UNIFORM DECOMPOSITION SPACES AND APPLICATIONS

Shaolei Ru

#### Abstract

We study multilinear operators $T(f_{1},f_{2},...,f_{m})$ that commutes with simultaneous translations and prove that if T is bounded from $L^{p_{1}} \times L^{p_{2}} \times ... \times L^{p_{m}}$ to $L^{p}$, then for any $r \geqslant p$, $0 \lt p,q \leqslant \infty$ and $s \gt \left\{ \begin{array}{lll} n(1-1\wedge\frac{1}{q}), &(\frac{1}{p},\frac{1}{q})\in D_{1};\\ n(1\vee\frac{1}{p}\vee\frac{1}{q}-\frac{1}{q}), &(\frac{1}{p},\frac{1}{q})\in \mathbb{R}_{+}^{2}-D_{1}, \end{array} \right.$ ($D_{1}=\{(\frac{1}{p},\frac{1}{q})\in\mathbb{R}_{+}^{2}:\frac{1}{q}\geqslant\frac{2}{p},\frac{1}{p}\leqslant\frac{1}{2}\}$)T is bounded from $M_{p_{1},q}^{s}\times M_{p_{2},q}^{s}\times...\times M_{p_{m},q}^{s}$ to $M_{r,q}^{s}$ (which improves the results obtained by [5], [6].), where $M_{p,q}^{s}$ is the modulation spaces. Besides, we alsoobtain the similar results for Triebel-type spaces $N_{p,q}^{s}$ introduced by [21] (T is bounded from $N_{p,q}^{s} \times N_{p,q}^{s} \times ... \times N_{p,q}^{s}$ to $N_{p,q}^{s}$). As applications, we obtain the boundedness on the modulation spaces for the bilinear Hilbert transform, bilinear fractional integral, the pointwise product of functions, and the bilinear oscillatory integral along parabolas. Also, in modulation spaces and $N_{p,q}^{s}$, we study the well-posedness of the Cauchy problem for the fractional heat and Schrödinger equations with some new nonlinear terms. Such nonlinear well-posedness problems are not studied in other function spaces.

#### Article information

Source
Taiwanese J. Math., Volume 18, Number 4 (2014), 1129-1149.

Dates
First available in Project Euclid: 10 July 2017

https://projecteuclid.org/euclid.twjm/1499706481

Digital Object Identifier
doi:10.11650/tjm.18.2014.3159

Mathematical Reviews number (MathSciNet)
MR3245434

Zentralblatt MATH identifier
1357.42029

#### Citation

Ru, Shaolei. MULTILINEAR ESTIMATES ON FREQUENCY-UNIFORM DECOMPOSITION SPACES AND APPLICATIONS. Taiwanese J. Math. 18 (2014), no. 4, 1129--1149. doi:10.11650/tjm.18.2014.3159. https://projecteuclid.org/euclid.twjm/1499706481

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